DOCSLIB.ORG

Theory of Computation

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

The Australian National University Semester 1, 2020 Research School of Computer Science Tutorial 12 Dirk Pattinson Theory of Computation Get into groups of two, grab a whiteboard marker each, and write your names on top of your side of the board. Exercises with a ! denote harder ones, !! denotes very difficult, and !!! denotes challenge exercises beyond the scope of the course. Exercise 1 Definitions PSPACE, or polynomially bounded space , is the class of languages recognized by a deterministic Turing machine, with the condition that the tape head never visits more than p(n) cells on the tape, where n is the length of the input, and p is some polynomial. BPP, or bounded-error probabilistic polynomial time, is the class of languages recognized by a randomized Turing machine, with the condition that • The probability of the machine making a error (accepting strings not in the language, or rejecting 1 strings in the language) is bounded above by 3 . • The worst case run time is bounded above by some polynomial p(n), where n is the length of input. 2 p(n) More formally, for an input string of length n, at least 3 of all binary strings of length 2 placed on 1 the random tape cause the machine to accept (if x 2 L), and at most 3 of all binary strings of length 2p(n) placed on the random tape cause the machine to accept (if x 62 L). EXPTIME, or exponential time, is the class of languages recognized by a deterministic Turing machine, where the worst case run time is bounded above by 2p(n), where p is a polynomial, and n is the length of the input. Exercise 2 Comparing classes Prove the following inclusions. 1.P ⊆ PSPACE 2.P ⊆ EXPTIME 3. ! NP ⊆ PSPACE 4. ! BPP ⊆ PSPACE 5. !! PSPACE ⊆ EXPTIME Exercise 3 PSPACE-Complete A language L is PSPACE-Hard if there exists a polytime reduction from L0 to L, where L0 is any problem in PSPACE. A language is PSPACE-Complete if it is both PSPACE and PSPACE-Hard. A quantified boolean formula is a boolean formula permitting quantifiers for variables. For example, 8x:9y:x _ y (1) and (9y:y ^ z) _:(8x:x) (2) A fully quantified boolean formula, or sentence, is a quantified boolean formula with no free variables, that is, every variable appears within the scope of a quantifier that refers to it. For example, (1) is a sentence, but (2) is not (as z is free). Note that every sentence must be either true or false, as there are no free variables. The quantified Boolean formula problem, or QBF, is the problem that, given a fully quantified boolean formula, is it equivalent to true? By an analogous proof to Cook's theorem, it can be shown that QBF is PSPACE-Complete. 1. Let an (alternating quantified boolean formula) denote a quantified boolean formula of the form 9x1:8x2:9x3 ::: 9xn.φ where each of the xi's represents any variable, and φ is a boolean expression. The alternating quantified Boolean formula problem, or AQBF, is the problem that, given a fully quantified alternating boolean formula, is it equivalent to true? Show that AQBF is PSPACE-Complete. The Geography game is a game played between two players. Given a finite directed graph with one vertex designated as a starting node, the players take turns listing nodes that form a simple path (no repeated verticies allowed.) The first person that cannot extend the path loses. Consider the following directed graph. 4 2 7 1 5 9 3 8 6 2. Player 1 goes first. Show that there is a winning strategy for Player 2. Formally, we define a geography graph G as a 3-tuple G = (V; E; s) where (V; E) forms a directed graph, jV j < 1, and s 2 V denotes the starting node. The geography problem (GEO) is as follows: Given a geography graph G, is there a strategy for Player 1 such that they can force a win? 3. ! Define the DAG-GEO problem to be the same as the GEO problem, but the graphs are restricted to be directed acyclic graphs. Prove that DAG-GEO is in P. 4. !! Show that GEO is PSPACE-complete. You may assume that 3-AQBF (an AQBF formula where the boolean expression inside is 3-CNF) is PSPACE-Complete. (Hint: Think about how you can convert a 3-AQBF formula with alternating quantifiers to having players alternating guessing assignments to those variables. Have player 1 handle the existential quantifiers, and player 2 handle the universal quantifiers. Consider the following gadget as a starting point.) s x1 :x1 x2 :x2 ::: Figure 1: Geography game gadget.
Recommended publications
  • CS601 DTIME and DSPACE Lecture 5 Time and Space Functions: T, S

    CS601 DTIME and DSPACE Lecture 5 Time and Space Functions: T, S

    CS601 DTIME and DSPACE Lecture 5 Time and Space functions: t, s : N → N+ Definition 5.1 A set A ⊆ U is in DTIME[t(n)] iff there exists a deterministic, multi-tape TM, M, and a constant c, such that, 1. A = L(M) ≡ w ∈ U M(w)=1 , and 2. ∀w ∈ U, M(w) halts within c · t(|w|) steps. Definition 5.2 A set A ⊆ U is in DSPACE[s(n)] iff there exists a deterministic, multi-tape TM, M, and a constant c, such that, 1. A = L(M), and 2. ∀w ∈ U, M(w) uses at most c · s(|w|) work-tape cells. (Input tape is “read-only” and not counted as space used.) Example: PALINDROMES ∈ DTIME[n], DSPACE[n]. In fact, PALINDROMES ∈ DSPACE[log n]. [Exercise] 1 CS601 F(DTIME) and F(DSPACE) Lecture 5 Definition 5.3 f : U → U is in F (DTIME[t(n)]) iff there exists a deterministic, multi-tape TM, M, and a constant c, such that, 1. f = M(·); 2. ∀w ∈ U, M(w) halts within c · t(|w|) steps; 3. |f(w)|≤|w|O(1), i.e., f is polynomially bounded. Definition 5.4 f : U → U is in F (DSPACE[s(n)]) iff there exists a deterministic, multi-tape TM, M, and a constant c, such that, 1. f = M(·); 2. ∀w ∈ U, M(w) uses at most c · s(|w|) work-tape cells; 3. |f(w)|≤|w|O(1), i.e., f is polynomially bounded. (Input tape is “read-only”; Output tape is “write-only”.
  • EXPSPACE-Hardness of Behavioural Equivalences of Succinct One

    EXPSPACE-Hardness of Behavioural Equivalences of Succinct One

    EXPSPACE-hardness of behavioural equivalences of succinct one-counter nets Petr Janˇcar1 Petr Osiˇcka1 Zdenˇek Sawa2 1Dept of Comp. Sci., Faculty of Science, Palack´yUniv. Olomouc, Czech Rep. [email protected], [email protected] 2Dept of Comp. Sci., FEI, Techn. Univ. Ostrava, Czech Rep. [email protected] Abstract We note that the remarkable EXPSPACE-hardness result in [G¨oller, Haase, Ouaknine, Worrell, ICALP 2010] ([GHOW10] for short) allows us to answer an open complexity ques- tion for simulation preorder of succinct one counter nets (i.e., one counter automata with no zero tests where counter increments and decrements are integers written in binary). This problem, as well as bisimulation equivalence, turn out to be EXPSPACE-complete. The technique of [GHOW10] was referred to by Hunter [RP 2015] for deriving EXPSPACE-hardness of reachability games on succinct one-counter nets. We first give a direct self-contained EXPSPACE-hardness proof for such reachability games (by adjust- ing a known PSPACE-hardness proof for emptiness of alternating finite automata with one-letter alphabet); then we reduce reachability games to (bi)simulation games by using a standard “defender-choice” technique. 1 Introduction arXiv:1801.01073v1 [cs.LO] 3 Jan 2018 We concentrate on our contribution, without giving a broader overview of the area here. A remarkable result by G¨oller, Haase, Ouaknine, Worrell [2] shows that model checking a fixed CTL formula on succinct one-counter automata (where counter increments and decre- ments are integers written in binary) is EXPSPACE-hard. Their proof is interesting and nontrivial, and uses two involved results from complexity theory.
  • Spring 2020 (Provide Proofs for All Answers) (120 Points Total)

    Spring 2020 (Provide Proofs for All Answers) (120 Points Total)

    Complexity Qual Spring 2020 (provide proofs for all answers) (120 Points Total) 1 Problem 1: Formal Languages (30 Points Total): For each of the following languages, state and prove whether: (i) the lan- guage is regular, (ii) the language is context-free but not regular, or (iii) the language is non-context-free. n n Problem 1a (10 Points): La = f0 1 jn ≥ 0g (that is, the set of all binary strings consisting of 0's and then 1's where the number of 0's and 1's are equal). ∗ Problem 1b (10 Points): Lb = fw 2 f0; 1g jw has an even number of 0's and at least two 1'sg (that is, the set of all binary strings with an even number, including zero, of 0's and at least two 1's). n 2n 3n Problem 1c (10 Points): Lc = f0 #0 #0 jn ≥ 0g (that is, the set of all strings of the form 0 repeated n times, followed by the # symbol, followed by 0 repeated 2n times, followed by the # symbol, followed by 0 repeated 3n times). Problem 2: NP-Hardness (30 Points Total): There are a set of n people who are the vertices of an undirected graph G. There's an undirected edge between two people if they are enemies. Problem 2a (15 Points): The people must be assigned to the seats around a single large circular table. There are exactly as many seats as people (both n). We would like to make sure that nobody ever sits next to one of their enemies.
  • Probabilistic Proof Systems: a Primer

    Probabilistic Proof Systems: a Primer

    Probabilistic Proof Systems: A Primer Oded Goldreich Department of Computer Science and Applied Mathematics Weizmann Institute of Science, Rehovot, Israel. June 30, 2008 Contents Preface 1 Conventions and Organization 3 1 Interactive Proof Systems 4 1.1 Motivation and Perspective ::::::::::::::::::::::: 4 1.1.1 A static object versus an interactive process :::::::::: 5 1.1.2 Prover and Veri¯er :::::::::::::::::::::::: 6 1.1.3 Completeness and Soundness :::::::::::::::::: 6 1.2 De¯nition ::::::::::::::::::::::::::::::::: 7 1.3 The Power of Interactive Proofs ::::::::::::::::::::: 9 1.3.1 A simple example :::::::::::::::::::::::: 9 1.3.2 The full power of interactive proofs ::::::::::::::: 11 1.4 Variants and ¯ner structure: an overview ::::::::::::::: 16 1.4.1 Arthur-Merlin games a.k.a public-coin proof systems ::::: 16 1.4.2 Interactive proof systems with two-sided error ::::::::: 16 1.4.3 A hierarchy of interactive proof systems :::::::::::: 17 1.4.4 Something completely di®erent ::::::::::::::::: 18 1.5 On computationally bounded provers: an overview :::::::::: 18 1.5.1 How powerful should the prover be? :::::::::::::: 19 1.5.2 Computational Soundness :::::::::::::::::::: 20 2 Zero-Knowledge Proof Systems 22 2.1 De¯nitional Issues :::::::::::::::::::::::::::: 23 2.1.1 A wider perspective: the simulation paradigm ::::::::: 23 2.1.2 The basic de¯nitions ::::::::::::::::::::::: 24 2.2 The Power of Zero-Knowledge :::::::::::::::::::::: 26 2.2.1 A simple example :::::::::::::::::::::::: 26 2.2.2 The full power of zero-knowledge proofs ::::::::::::
  • A Short History of Computational Complexity

    A Short History of Computational Complexity

    The Computational Complexity Column by Lance FORTNOW NEC Laboratories America 4 Independence Way, Princeton, NJ 08540, USA [email protected] http://www.neci.nj.nec.com/homepages/fortnow/beatcs Every third year the Conference on Computational Complexity is held in Europe and this summer the University of Aarhus (Denmark) will host the meeting July 7-10. More details at the conference web page http://www.computationalcomplexity.org This month we present a historical view of computational complexity written by Steve Homer and myself. This is a preliminary version of a chapter to be included in an upcoming North-Holland Handbook of the History of Mathematical Logic edited by Dirk van Dalen, John Dawson and Aki Kanamori. A Short History of Computational Complexity Lance Fortnow1 Steve Homer2 NEC Research Institute Computer Science Department 4 Independence Way Boston University Princeton, NJ 08540 111 Cummington Street Boston, MA 02215 1 Introduction It all started with a machine. In 1936, Turing developed his theoretical com- putational model. He based his model on how he perceived mathematicians think. As digital computers were developed in the 40's and 50's, the Turing machine proved itself as the right theoretical model for computation. Quickly though we discovered that the basic Turing machine model fails to account for the amount of time or memory needed by a computer, a critical issue today but even more so in those early days of computing. The key idea to measure time and space as a function of the length of the input came in the early 1960's by Hartmanis and Stearns.
  • 26 Space Complexity

    26 Space Complexity

    CS:4330 Theory of Computation Spring 2018 Computability Theory Space Complexity Haniel Barbosa Readings for this lecture Chapter 8 of [Sipser 1996], 3rd edition. Sections 8.1, 8.2, and 8.3. Space Complexity B We now consider the complexity of computational problems in terms of the amount of space, or memory, they require B Time and space are two of the most important considerations when we seek practical solutions to most problems B Space complexity shares many of the features of time complexity B It serves a further way of classifying problems according to their computational difficulty 1 / 22 Space Complexity Definition Let M be a deterministic Turing machine, DTM, that halts on all inputs. The space complexity of M is the function f : N ! N, where f (n) is the maximum number of tape cells that M scans on any input of length n. Definition If M is a nondeterministic Turing machine, NTM, wherein all branches of its computation halt on all inputs, we define the space complexity of M, f (n), to be the maximum number of tape cells that M scans on any branch of its computation for any input of length n. 2 / 22 Estimation of space complexity Let f : N ! N be a function. The space complexity classes, SPACE(f (n)) and NSPACE(f (n)), are defined by: B SPACE(f (n)) = fL j L is a language decided by an O(f (n)) space DTMg B NSPACE(f (n)) = fL j L is a language decided by an O(f (n)) space NTMg 3 / 22 Example SAT can be solved with the linear space algorithm M1: M1 =“On input h'i, where ' is a Boolean formula: 1.
  • Introduction to Complexity Theory Big O Notation Review Linear Function: R(N)=O(N)

    Introduction to Complexity Theory Big O Notation Review Linear Function: R(N)=O(N)

    GS019 - Lecture 1 on Complexity Theory Jarod Alper (jalper) Introduction to Complexity Theory Big O Notation Review Linear function: r(n)=O(n). Polynomial function: r(n)=2O(1) Exponential function: r(n)=2nO(1) Logarithmic function: r(n)=O(log n) Poly-log function: r(n)=logO(1) n Definition 1 (TIME) Let t : . Define the time complexity class, TIME(t(n)) to be ℵ−→ℵ TIME(t(n)) = L DTM M which decides L in time O(t(n)) . { |∃ } Definition 2 (NTIME) Let t : . Define the time complexity class, NTIME(t(n)) to be ℵ−→ℵ NTIME(t(n)) = L NDTM M which decides L in time O(t(n)) . { |∃ } Example 1 Consider the language A = 0k1k k 0 . { | ≥ } Is A TIME(n2)? Is A ∈ TIME(n log n)? Is A ∈ TIME(n)? Could∈ we potentially place A in a smaller complexity class if we consider other computational models? Theorem 1 If t(n) n, then every t(n) time multitape Turing machine has an equivalent O(t2(n)) time single-tape turing≥ machine. Proof: see Theorem 7:8 in Sipser (pg. 232) Theorem 2 If t(n) n, then every t(n) time RAM machine has an equivalent O(t3(n)) time multi-tape turing machine.≥ Proof: optional exercise Conclusion: Linear time is model specific; polynomical time is model indepedent. Definition 3 (The Class P ) k P = [ TIME(n ) k Definition 4 (The Class NP) GS019 - Lecture 1 on Complexity Theory Jarod Alper (jalper) k NP = [ NTIME(n ) k Equivalent Definition of NP NP = L L has a polynomial time verifier .
  • Is It Easier to Prove Theorems That Are Guaranteed to Be True?

    Is It Easier to Prove Theorems That Are Guaranteed to Be True?

    Is it Easier to Prove Theorems that are Guaranteed to be True? Rafael Pass∗ Muthuramakrishnan Venkitasubramaniamy Cornell Tech University of Rochester [email protected] [email protected] April 15, 2020 Abstract Consider the following two fundamental open problems in complexity theory: • Does a hard-on-average language in NP imply the existence of one-way functions? • Does a hard-on-average language in NP imply a hard-on-average problem in TFNP (i.e., the class of total NP search problem)? Our main result is that the answer to (at least) one of these questions is yes. Both one-way functions and problems in TFNP can be interpreted as promise-true distri- butional NP search problems|namely, distributional search problems where the sampler only samples true statements. As a direct corollary of the above result, we thus get that the existence of a hard-on-average distributional NP search problem implies a hard-on-average promise-true distributional NP search problem. In other words, It is no easier to find witnesses (a.k.a. proofs) for efficiently-sampled statements (theorems) that are guaranteed to be true. This result follows from a more general study of interactive puzzles|a generalization of average-case hardness in NP|and in particular, a novel round-collapse theorem for computationally- sound protocols, analogous to Babai-Moran's celebrated round-collapse theorem for information- theoretically sound protocols. As another consequence of this treatment, we show that the existence of O(1)-round public-coin non-trivial arguments (i.e., argument systems that are not proofs) imply the existence of a hard-on-average problem in NP=poly.
  • On Exponential-Time Completeness of the Circularity Problem for Attribute Grammars

    On Exponential-Time Completeness of the Circularity Problem for Attribute Grammars

    On Exponential-Time Completeness of the Circularity Problem for Attribute Grammars Pei-Chi Wu Department of Computer Science and Information Engineering National Penghu Institute of Technology Penghu, Taiwan, R.O.C. E-mail: [email protected] Abstract Attribute grammars (AGs) are a formal technique for defining semantics of programming languages. Existing complexity proofs on the circularity problem of AGs are based on automata theory, such as writing pushdown acceptor and alternating Turing machines. They reduced the acceptance problems of above automata, which are exponential-time (EXPTIME) complete, to the AG circularity problem. These proofs thus show that the circularity problem is EXPTIME-hard, at least as hard as the most difficult problems in EXPTIME. However, none has given a proof for the EXPTIME- completeness of the problem. This paper first presents an alternating Turing machine for the circularity problem. The alternating Turing machine requires polynomial space. Thus, the circularity problem is in EXPTIME and is then EXPTIME-complete. Categories and Subject Descriptors: D.3.1 [Programming Languages]: Formal Definitions and Theory ¾ semantics; D.3.4 [Programming Languages]: Processors ¾ compilers, translator writing systems and compiler generators; F.2.2 [Analysis of Algorithms and Problem Complexity] Nonnumerical Algorithms and Problems ¾ Computations on discrete structures; F.4.2 [Grammars and Other Rewriting Systems] decision problems Key Words: attribute grammars, alternating Turing machines, circularity problem, EXPTIME-complete. 1. INTRODUCTION Attribute grammars (AGs) [8] are a formal technique for defining semantics of programming languages. There are many AG systems (e.g., Eli [4] and Synthesizer Generator [10]) developed to assist the development of “language processors”, such as compilers, semantic checkers, language-based editors, etc.
  • Computational Complexity

    Computational Complexity

    Computational complexity Plan Complexity of a computation. Complexity classes DTIME(T (n)). Relations between complexity classes. Complete problems. Domino problems. NP-completeness. Complete problems for other classes Alternating machines. – p.1/17 Complexity of a computation A machine M is T (n) time bounded if for every n and word w of size n, every computation of M has length at most T (n). A machine M is S(n) space bounded if for every n and word w of size n, every computation of M uses at most S(n) cells of the working tape. Fact: If M is time or space bounded then L(M) is recursive. If L is recursive then there is a time and space bounded machine recognizing L. DTIME(T (n)) = fL(M) : M is det. and T (n) time boundedg NTIME(T (n)) = fL(M) : M is T (n) time boundedg DSPACE(S(n)) = fL(M) : M is det. and S(n) space boundedg NSPACE(S(n)) = fL(M) : M is S(n) space boundedg . – p.2/17 Hierarchy theorems A function S(n) is space constructible iff there is S(n)-bounded machine that given w writes aS(jwj) on the tape and stops. A function T (n) is time constructible iff there is a machine that on a word of size n makes exactly T (n) steps. Thm: Let S2(n) be a space-constructible function and let S1(n) ≥ log(n). If S1(n) lim infn!1 = 0 S2(n) then DSPACE(S2(n)) − DSPACE(S1(n)) is nonempty. Thm: Let T2(n) be a time-constructible function and let T1(n) log(T1(n)) lim infn!1 = 0 T2(n) then DTIME(T2(n)) − DTIME(T1(n)) is nonempty.
  • ATL* Satisfiability Is 2EXPTIME-Complete*

    ATL* Satisfiability Is 2EXPTIME-Complete*

    ATL* Satisfiability is 2EXPTIME-Complete⋆ Sven Schewe Universit¨at des Saarlandes, 66123 Saarbr¨ucken, Germany Abstract. The two central decision problems that arise during the de- sign of safety critical systems are the satisfiability and the model checking problem. While model checking can only be applied after implementing the system, satisfiability checking answers the question whether a system that satisfies the specification exists. Model checking is traditionally con- sidered to be the simpler problem – for branching-time and fixed point logics such as CTL, CTL*, ATL, and the classical and alternating time µ-calculus, the complexity of satisfiability checking is considerably higher than the model checking complexity. We show that ATL* is a notable exception of this rule: Both ATL* model checking and ATL* satisfiability checking are 2EXPTIME-complete. 1 Introduction One of the main challenges in system design is the construction of correct im- plementations from their temporal specifications. Traditionally, system design consists of three separated phases, the specification phase, the implementation phase, and the validation phase. From a scientific point of view it seems inviting to overcome the separation between the implementation and validation phase, and replace the manual implementation of a system and its subsequent validation by a push-button approach, which automatically synthesizes an implementation that is correct by construction. Automating the system construction also pro- vides valuable additional information: we can distinguish unsatisfiable system specifications, which otherwise would go unnoticed, leading to a waste of effort in the fruitless attempt of finding a correct implementation. One important choice on the way towards the ideal of fully automated system construction is the choice of the specification language.
  • NP-Complete Problems

    NP-Complete Problems

    NP-complete problems • Informally, these are the hardest problems in the class NP CS 3813: Introduction to Formal • If any NP-complete problem can be solved by a Languages and Automata polynomial time deterministic algorithm, then every problem in NP can be solved by a polynomial time deterministic algorithm NP-Completeness • But no polynomial time deterministic algorithm is known to solve any of them Examples of NP-complete problems Polynomial-time reduction • Traveling salesman problem Let L1 and L2 be two languages over alphabets • Hamiltonian cycle problem Σ1 and Σ2,, respectively. L1 is said to be • Clique problem polynomial-time reducible to L2 if there is a • Subset sum problem total function f: Σ1*→Σ2* for which • Boolean satisfiability problem 1) x ∈ L1 if an only if f(x) ∈ L2, and 2) f can be computed in polynomial time • Many thousands of other important The function f is called a polynomial-time computational problems in computer science, reduction. mathematics, economics, manufacturing, communications, etc. Another view Exercise In English, describe a Turing machine that Polynomial-time reduction reduces L1 to L2 in polynomial time. Using Set of instances of Set of instances of big-O notation, give the time complexity of Problem Q Problem Q’ the machine that computes the reduction. –L = {aibiaj}L= {cidi} One decision problem is polynomial-time reducible to 1 2 i i i i another if a polynomial time algorithm can be developed –L1 = {a (bb) }L2 = {a b } that changes each instance of the first problem to an instance of the second such that a yes (or no) answer to the second problem entails a yes (or no) answer to the first.